Introduction

In earlier chapters it was shown that Brownian motion and colloidal interparticle forces give rise to viscoelastic effects. When a constant shear rate is applied to some colloidal suspensions, the viscosity can exhibit long transients, while viscoelastic features such as normal stress differences are hardly detectable. A well-known daily life example is provided by tomato ketchup: shaking turns it from a gel-like substance into a free-flowing liquid, but when left alone it will gradually stiffen and return to a gel. This is an example of the more general phenomenon known as thixotropy. It has been reported for a large number of colloidal products, some of which are listed in Table 7.1. They are most often colloidal glasses or gels at rest. Extensive lists of thixotropic products can be found in the literature [1–4]. Some products are actually formulated to exhibit a well-defined time evolution for viscosity recovery after shearing. Special additives (“thixotropic agents”) are available to induce and control such behavior. We note that many complex fluids such as some polymeric systems, liquid crystals and micellar systems exhibit thixotropy; however, these interesting materials are beyond our scope.

There is an extensive body of papers on thixotropy, scattered over the scientific and technical literature, including some reviews [1–5]. Nevertheless, the subject has been essentially ignored in rational continuum mechanics and, until recently, in colloid science. An explanation can perhaps be found in the persistent ambiguity about its definition, the lack of suitable model systems for study, and the complexity of the phenomenon, which includes serious measurement challenges. The more recent interest in glasses and gels within the general area of soft condensed matter is providing new terminology in the field, such as aging and shear rejuvenation [6]. This chapter provides a guide to understanding thixotropy in colloidal suspensions and an introduction to its modeling.

The subject of this book is the rheology of colloidal and nanoparticle dispersions. The reader will quickly appreciate the breadth of the subject area and, furthermore, that mastering colloidal suspension rheology requires some basic knowledge in colloid science as well as rheology. Thus, this chapter introduces some basic and simplified concepts in colloid science and rheology prior to embarking on the main theme of the book. As the term colloid is very general we necessarily need to focus on fundamental aspects of basic colloidal particles, their interactions, and their dispersion thermodynamic properties. These are, of course, the basis for understanding more complex systems. The rheology section is provided as an introduction to the basic concepts (a more advanced treatment of rheological testing of colloidal dispersions is provided in Chapter 9). Therefore, this chapter provides the minimum level of understanding that the reader will find valuable for understanding colloidal suspension rheology, as well as a means to introduce nomenclature and concepts used throughout the book. As a consequence, a reader familiar with either or both subjects may still find it valuable to skim through the material or refer back to it as needed.

Colloidal phenomena

Colloid science is a rich field with an equally rich literature. The reader is referred to a number of excellent monographs that cover the basics of colloid science in much greater detail. These will be presented without derivation. In particular, we use nomenclature and presentation of many ideas following Colloidal Dispersions [1] and Principles of Colloid and Surface Chemistry [2], which may be of help for further reading and inquiry, and for derivations of the results presented herein. Indeed, there are many additional excellent textbooks and monographs on colloid science, and references are provided where they are most relevant throughout this chapter as well as in the other chapters.

Rationale and organization

Models for the turbulent stresses and scalar fluxes have been in widespread use since the 1960s, incorporated within CFD codes of a wide range of types and capabilities. Over this period the vast majority of computations have been made using turbulence models simpler than second-moment closure. Quite clearly, such simpler models must deliver satisfactory predictions of some of the flows of interest – for otherwise they would be discarded. This chapter is devoted to such reduced models. The position adopted is that, of course, such simplification makes sense, provided it is made with an appreciation of what has been lost in the process.

This truism applies as much to the numerical solver as to the physical model of turbulence employed, for one would surely never use a three-dimensional, elliptic, compressible-flow solver if one's interests were simply in computing a range of axisymmetric, unseparating boundary layers in liquids. But, if we proceed in the reverse direction, while it is not usually possible to apply a simple numerical solver to flows well beyond the solver's capability, it is all too easy to assume that a turbulence model that functioned very satisfactorily in computing simple shear flows, will perform equally as well in computing complex strains or in the presence of strong external force fields. That is why it is seen as important that simple (or simpler) turbulence models should be arrived at by a rational simplification of the full second-moment closure (having regard for the particular features of the flow to be computed) rather than by adopting some constitutive equation as an article of faith.

Early proposals

The label wall functions was first applied by Patankar and Spalding (1967) as the collective name for the set of algebraic relations linking the values of the effective wall-normal gradients of dependent variables between the wall and the wall-adjacent node (in a numerical solver) to the shear stress, heat or mass flux at the wall.

The underlying purpose of wall functions, as originally proposed, was to allow computations to escape the need to model the very complex flow dynamics associated with the low-Re region that formed the subject of Chapter 6. It may seem absurd that in the region which, from a physical point of view, contains the most complex viscous and turbulent interactions, one resorts to algebraic rather than differential relations to resolve the flow. We note, however, that in Chapter 7 the power of using very simple eddy-viscosity models of turbulence to handle the sublayer has been demonstrated. Wall functions may just be seen as an extrapolation of that simplification strategy; that is, an even cheaper approach to capturing the essentials of the viscosity-affected layer, by exploiting the fact that gradients of dependent variables normal to the wall are dominant and that transport effects are relatively uninfluential. The present chapter first summarizes conventional wall functions and then introduces four more powerful approaches that the authors and their colleagues have developed more recently.

The fact of turbulent flow

Man has evolved within a world where air and water are, by far, the most common fluids encountered. The scales of the environment around him and of the machines and artefacts his ingenuity has created mean that, given their relatively low kinematic viscosities, the relevant global Reynolds number, Re, associated with the motion of both fluids is, in most cases, sufficiently high that the resultant flow is of the continually time-varying, spatially irregular kind we call turbulent.

If, however, our Reynolds number is chosen not by the overall physical dimension of the body of interest – an aircraft wing, say – and the fluid velocity past it but by the smallest distance over which the velocity found within a turbulent eddy changes appreciably and the time over which such a velocity change will occur, its value then turns out to be of order unity. Indeed, one might observe that if this last Reynolds number, traditionally called the micro-scale Reynolds number, Reη, were significantly greater than unity, the rate at which the turbulent kinetic energy is destroyed by viscous dissipation could not balance the overall rate at which turbulence ‘captures’ kinetic energy from the mean flow.

The nature of viscous and wall effects: options for modelling

The turbulence models considered in earlier chapters were based on the assumption that the turbulent Reynolds numbers were high enough everywhere to permit the neglect of viscous effects. Thus, they are not applicable to flows with a low bulk Reynolds number (where the effects of viscosity may permeate the whole flow) or to the viscosity-affected regions adjacent to solid walls (commonly referred to as the viscous sublayer and buffer regions but which we shall normally collectively refer to as the viscous region) which always exist on a smooth wall irrespective of how high the bulk Reynolds number may be. In other words, while at high Reynolds number, viscous effects on the energy-containing turbulent motions are indeed negligible throughout most of the flow, the condition of no-slip at solid interfaces always ensures that, in the immediate vicinity of a wall, viscous contributions will be influential, perhaps dominant. Figure 6.1 shows the typical ‘layered’ composition for a near-wall turbulent flow (though with an expanded scale for the near-wall region) as found in a constant-pressure boundary layer, channel or pipe flow. Although the thickness of this viscosity-affected zone is usually two or more orders of magnitude less than the overall width of the flow (and decreases as the Reynolds number increases), its effects extend over the whole flow field since, typically, half of the velocity change from the wall to the free stream occurs in this region.

Because viscosity dampens velocity fluctuations equally in all directions, one may argue that viscosity has a ‘scalar’ effect. However, turbulence in the proximity of a solid wall or a phase interface is also subjected to non-viscous damping arising from the impermeability of the wall and the consequent reflection of pressure fluctuations. This ‘wall-blocking’ effect, which is also felt outside the viscous layer well into the fully turbulent wall region, directly dampens the velocity fluctuations in the wall-normal direction and thus it has a ‘vector’ character. A good illustration of this effect is the reduction of the surface-normal velocity fluctuations that has been observed in flow regions close to a phase interface, where there are no viscous effects, for example the DNS of Perot and Moin (1995).

Scientific papers on how to represent in mathematical form the types of fluid motion we call turbulent flow have been appearing for over a century while, for the last sixty years or so, a sufficient body of knowledge has been accumulated to tempt a succession of authors to collect, systematize and distil a proportion of that knowledge into textbooks. From the start a bewildering variety of approaches has been advocated: thus, even in the 1970s, the algebraic mixing-length models presented in the book by Cebeci and Smith jostled on the book-shelves with Leslie's manful attempt to make comprehensible to a less specialized readership the direct-interaction approach developed by Kraichnan and colleagues. As the progressive advance in computing power made it possible to apply the emerging strategy of computational fluid dynamics to an ever-widening array of industrially important flows, however, eddy-viscosity models (EVMs) based on the solution of two transport equations for scalar properties of turbulence (essentially, length and time scales of the energy-containing eddies) emerged as the modelling strategy of choice and, correspondingly, have been the principal focus in several textbooks on the modelling of turbulent flows (for example, Launder and Spalding, Wilcox and Piquet).

Today, two-equation EVMs remain the work-horse of industrial CFD and are applied through commercially marketed software to flows of a quite bewildering complexity, though often with uncertain accuracy. However, there has been a major shift among the modelling research community to abandon approaches based on the Reynolds-averaged Navier–Stokes (RANS) equations in favour of large-eddy simulation (LES) where the numerical solution for any flow adopts a three-dimensional, time-dependent discretization of the Navier–Stokes equations using a model to account simply for the effects of turbulent motions too fine in scale to be resolved with the mesh adopted – that is, a sub-grid-scale (or sgs) model. While acknowledging that LES offers the prospects of tackling turbulence problems beyond the scope of RANS, a further major driver for this changeover has been the manifold inadequacies of the stress-strain hypothesis adopted by linear eddy-viscosity models. While such a simple linkage between mean strain rate and turbulent stress seemed adequate for a large proportion of two-dimensional, nearly parallel flows, its weaknesses became abundantly clear as attention shifted to recirculating, impinging and three-dimensional shear flows. Although an LES approach will, most probably, also adopt an sgs model of eddy-viscosity type, the consequences are less serious for two reasons. First, the majority of the transport caused by the turbulent motion will be directly resolved by the large eddies and secondly, the finer scale eddies that must still be resolved by the sub-grid-scale model of turbulence will arguably be a good deal closer to isotropy. Thus, adopting an isotropic eddy viscosity as the sgs model may not significantly impair the accuracy of the solution.